Freezing causes a Pokémon to be unable to make a move for an indeterminate number of moves.
The specifics vary between generations.

Generation I
Once frozen, a Pokémon can't be thawed out in battle other than via the use of items such as an Ice Heal or being hit by a damaging Fire-type move
.
Generation II onwards
A frozen Pokémon has a 20% chance of being thawed out every turn. On the turn that a Pokémon defrosts, it cannot attack until the next turn, similar to Sleep in Generation I. In addition, several moves can still be used by the frozen Pokémon while frozen (thawing it out in the process); Pokémon can't be frozen in intense sunlight, but they don't thaw out faster if already frozen, contrary to popular belief

Generation III onwards
Same as Generation II, but Pokémon can attack on the turn that they defrost. Animation also shows up on every turn Pokémon is frozen (in contrast to previous generations, where animation showed up only when freeze was inflicted).
Freezing also disables the Ability Flash Fire.

Generation IV onwards
Same as Generation III. A frozen Sky Forme Shaymin will revert to its Land Forme.
Generation V onwards
Pokémon glow blue and have their animation stopped when inflicted with freeze. It no longer stops Flash Fire from working.